COLLECTIVE EXCITATIONS IN ALPHA-HELICAL PROTEIN MOLECULE

Continuous energy supply, a necessary condition for life, excites a state far from thermodynamic equilibrium, in particular coherent electric polar vibrations depending on water ordering in the cell. The collective, low-frequency vibrational excitations in protein macromolecules in the terahertz frequency region are suggested to orchestrate many protein processes such as enzymatic activity, electron/energy transport, conformation transitions and others. The most significant type of excitations is long-lived coherent vibronic modes and waves which are either spreading over large protein subdomains or being localised states. Two possible mechanisms of the formation of collective dynamic modes in the form of Fröhlich collective mode and Davydov soliton were previously suggested. We developed a unified quantum-mechanics approach to describe conditions of the formation of Fröhlich vibronic state and Davidov soliton in alphahelical protein molecules interacting with the environment. We distinguish three subsystems in the model, i.e., (i) oscillating peptide groups (PGs), interacting with (ii) the subsystem of side residuals of proteins, which in turn interacts with the environment (surrounding water), which is responsible for dissipation and fluctuation processes and modelled by a system of harmonic oscillators. It was shown that the equation of motion for dynamic variables of PG chain (phonon variables) can be transformed to the nonlinear Schrodinger equation for order parameters, which determines energy “pumping” due to protein interaction with a reservoir. Solution of the order parameter equation was shown to admit bifurcation into the solution corresponding to the formation of weak damped collective vibronic mode with growing amplitude. It was also shown that the solution corresponding to Davydov soliton can exist at certain boundary conditions in this bifurcation region. The suggested mechanism of emergent of macroscopic dissipative structures in the form of collective vibronic modes in proteins is discussed in connection with recent experimental data on long-lived collective protein excitation in the terahertz frequency region

alpha-helical proteinα-helical protein, Fröhlich condensation, autolocalization states, hydrogen bonds, phonon, dynamical regimes, soliton

1. Fröhlich Н. The biological effects of microwaves and related questions //Advances in Electronics and Electron Physics. 1980. V. 53. P. 85–152.

2. Fröhlich H. Coherent electric vibrations in biological systems and cancer problem // IEEE Trans. Microwave Theory Tech. 1978. V. 26. Р. 613–617.

3. Warburg O., Posener K., Negelein E. Über den Stoffwechsel der Carcinomzelle // Biochem. Z. 1924. V. 152. Р. 309–344. 4. Warburg O. On the origin of cancer cells // Science. 1956. V. 123. Р. 309–314.

4. Pokorny J. Conditions for coherent vibrations in the cytoskeleton // Bioelectrochem. Bioenerg. 1999. V. 48(2). Р. 267–271.

5. Turton D.A., Senn H.M., Harwood T., Lapthorn A.J., Ellis E.M., Wynne K. Terahertz underdamped vibrational motion governs protein-ligand binding in solution // Nat. Commun. 2014. V. 5. 3999. doi:10.1038/ncomms4999.

6. Pokorný J., Pokorný J., Foletti A., Kobilková J., Vrba J., Vrba J. Mitochondrial dysfunction and disturbed coherence: gate to cancer // Pharmaceuticals. 2015. V. 8. P. 675–695.

7. Давыдов А.С. Солитоны в молекулярных системах. Киев: Наукова думка, 1984. 288 с.

8. Davydov A.S. Solitons in molecular systems. Kiev: Naukova dumka, 1984. 288 p. (in Russ.)

9. Takeno S. Vibron solitons in one-dimensional molecular crystals // Prog. Theor. Phys. 1984. V. 71. № 2. P. 395–398.

10. Lupichev L.N., Savin A.V., Kadantsev V.N. Synergetics of Molecular Systems. Springer Int. Publ. Switzerland, 2015. VIII. 332 p.

11. Lundholm I.V., Rodilla H., Wahlgren W., Duelli A., Bourenkov G., Vukusic J., Friedman R., Stake J., Schneider T. Terahertz radiation induces non-thermal structural changes, associated with Fröhlich condensation in a protein crystal // Structural Dynamics. 2015. V. 2. P. 054702.

12. Saint-Blancard J., Clochard A., Cozzone P., Berthou J., Jolles P., The temperaturedependent structural transition of lysozyme A study of the Arrhenius plots // Biochim. Biophys. Acta. 1977. V. 491(1). P. 354–356.

13. Borgia A., Borgia M.B., Bugge K., Kissling V.M., Heidarsson P.O., Fernandes C.B., Sottini A., Soranno A., Buholzer K.J., Nettels D., Kragelund B.B., Best R.B., Schuler B. Extreme disorder in an ultrahigh-affinity protein complex // Nature. 2018. March 1. V. 555. P. 61–66.

14.